線性拋物偏微分方程

Abstract

這篇論文中我們研究線性拋物偏微分方程。首先，我們舉出關於此種方程一些實際的例子。接下來，我們運用一般的方法去解決此方程，雖然同一例題用不同的解法，會得到不同的表示方式，但可以證明都是一樣的。 當我們應用Fourier和Laplace 轉換去解決全線和半線的偏微分方程時，我們必須使用inverse Fourier和Laplace 轉換去求出解析解，而這些被積分函數中有時會牽扯到平方根，但是在複數平面中，平方根是多值的。為了使我們的運算正確，所以我們從複數平面上的代數結構去發展黎曼空間，讓平方根是一個單值函數，我們可以利用數學軟體去完成inverse轉換。在此篇論文中，我們提出一些例題去說明及驗證此方法是可行的。

We study the linear parabolic partial differential equations(linear parabolic PDEs). First, we give some practical examples and show that they are governed by such type of the equations. Next, we apply several classical methods to solve the linear parabolic PDEs with the solutions being expressed in various forms. We then identify those solutions. When we apply Fourier and Laplace transformations to the whole-and half-line PDEs, it is necessary to perform the inverse Fourier and Laplace transformations to derive the PDE solutions, and it is quite often that those integrals involve the square root operator which is multi-valued in the complex plane. In order to perform the inverse transformations correctly, we develop the Riemann surfaces from the complex plane with the proper algebraic structures to assure that the square root is now a single-valued function on the surfaces, and we are able to accomplish the inverse transformations analytically and numerically. Some examples are given to illustrate the entire scheme.